Question Number 218438 by Marzuk last updated on 10/Apr/25 $${An}\:{amazing}\:{thing}\:{i}\:{saw} \\ $$$${S}\:=\:\mathrm{1}\:+\:\mathrm{2}\:+\:\mathrm{3}\:+\:\mathrm{4}\:+\:\mathrm{5}\:+\:\mathrm{6}… \\ $$$$\:\:\:\:=\:\mathrm{1}\:+\:\mathrm{2}\left(\mathrm{2}/\mathrm{2}\:+\:\mathrm{3}/\mathrm{2}\:+\:\mathrm{4}/\mathrm{2}\:+\:\mathrm{5}/\mathrm{2}\:+\mathrm{6}/\mathrm{2}….\right) \\ $$$$\:\:\:\:=\:\mathrm{1}\:+\:\mathrm{2}\left(\mathrm{1}\:+\:\mathrm{3}/\mathrm{2}\:+\:\mathrm{2}\:+\:\mathrm{5}/\mathrm{2}\:+\:\mathrm{3}…\right) \\ $$$$\:\:\:\:=\:\mathrm{1}\:+\:\mathrm{2}\left(\mathrm{1}+\:\mathrm{2}\:+\:\mathrm{3}\:…\:+\:\mathrm{3}/\mathrm{2}\:+\:\mathrm{5}/\mathrm{2}…\right) \\ $$$$\:\:\:\:=\:\mathrm{1}\:+\:\mathrm{2}{S}\:+\:\mathrm{2}\underset{{n}=\:\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{2}{n}\:+\:\mathrm{1}}{\mathrm{2}} \\ $$$${or},{S}\:−\:\mathrm{2}{S}\:=\:\mathrm{1}\:+\:\underset{{n}=\mathrm{1}} {\overset{\infty}…
Question Number 218208 by Marzuk last updated on 01/Apr/25 $${This}\:{question}\:{is}\:{really}\:{important} \\ $$$${Prove}\:{or}\:{disprove}\:{that} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{3}^{{n}} {m}+\mathrm{3}^{{n}−\mathrm{1}} }{\mathrm{2}^{\lceil\frac{{n}}{\mathrm{2}}\rceil} }\:+\:\frac{\mathrm{3}^{{n}−\mathrm{1}} }{\mathrm{2}^{{n}} }\: \\ $$$$\:{the}\:{limit}\:{exists}\:{for}\:{m}\:\in\:{N}\:\backslash{B} \\ $$$${where}\:{B}\:=\:\left\{{n}\:\mid\:{log}_{\mathrm{2}} \left({n}\right)\:\in\:{N}\:\right\}…
Question Number 218076 by malwan last updated on 28/Mar/25 $${How}\:{many}\:{ways}\:{to}\:{arrnge} \\ $$$${the}\:{letters}\:{ABCCCDEFG} \\ $$$$\left(\mathrm{1}\right)\:{in}\:{general}\:. \\ $$$$\left(\mathrm{2}\right)\:{all}\:\mathrm{3}\:{Cs}\:{must}\:{be}\:{together} \\ $$$$\left(\mathrm{3}\right)\:{only}\:\mathrm{2}\:{Cs}\:{must}\:{be}\:{together} \\ $$$$\left(\mathrm{4}\right)\:{no}\:\mathrm{2}\:{or}\:\mathrm{3}\:{Cs}\:{be}\:{together} \\ $$$$\left(\mathrm{5}\right)\:{no}\:{letter}\:{still}\:\:{in}\:{its} \\ $$$${original}\:{place}\:. \\…
Question Number 217958 by dscm last updated on 23/Mar/25 $${a},{b},{c}\in\mathbb{Z}^{+} {and} \\ $$$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +{ab}+{bc}+{ca}=\mathrm{2025} \\ $$$${find}\:{out}\:{all}\:{triplets}\:\left({a},{b},{c}\right). \\ $$ Commented by Unhombre last updated…
Question Number 217952 by dscm last updated on 23/Mar/25 $${If}\:{x}\in\mathbb{Z}\:\wedge{y}\:{non}-{negative}\:{integer} \\ $$$${such}\:{that} \\ $$$${x}^{\mathrm{2}} +\mathrm{10}{x}+\mathrm{23}=\mathrm{2}^{{y}} \\ $$$${find}\:{out}\:{x},{y} \\ $$ Answered by Frix last updated on…
Question Number 217244 by ArshadS last updated on 07/Mar/25 $$\:\mathrm{Find}\:\mathrm{all}\:\mathrm{two}-\mathrm{digit}\:\mathrm{numbers}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{when}\:\mathrm{the}\:\mathrm{number}\:\mathrm{is}\:\mathrm{divided}\:\mathrm{by} \\ $$$$\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{its}\:\mathrm{digits}\:\mathrm{the}\:\mathrm{quotient}\: \\ $$$$\mathrm{is}\:\mathrm{4}\:\mathrm{and}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{is}\:\mathrm{3}. \\ $$ Answered by Frix last updated on 07/Mar/25…
Question Number 217132 by ArshadS last updated on 02/Mar/25 $$ \\ $$$$\mathrm{Find}\:\mathrm{all}\:\mathrm{integers}\:\:\mathrm{n}>\:\mathrm{1}\:\mathrm{such}\:\mathrm{that} \\ $$$$\:\:\mathrm{n}\:\:\mathrm{divides}\:\:\mathrm{2}^{\mathrm{n}−\mathrm{1}} \:+\:\mathrm{3}^{\mathrm{n}−\mathrm{1}} . \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 217129 by ArshadS last updated on 02/Mar/25 $${prove}\:{that}\:{if}\:{an}\:{integer}\:{n}\:{is}\:{not}\:{divisible}\:{by}\:\mathrm{2}\:{or}\:\mathrm{3} \\ $$$$\:{then}\:{n}^{\mathrm{2}} \equiv\mathrm{1}\left({mod}\:\mathrm{24}\right) \\ $$ Commented by A5T last updated on 02/Mar/25 $$\mathrm{This}\:\mathrm{is}\:\mathrm{not}\:\mathrm{necessarily}\:\mathrm{true}.\: \\ $$$$\mathrm{n}\:\mathrm{could}\:\mathrm{also}\:\mathrm{be}\:\equiv\:\mathrm{5},\mathrm{7},\mathrm{11},\mathrm{13},\mathrm{17},\mathrm{19},\mathrm{23}\:\left(\mathrm{mod}\:\mathrm{24}\right)…
Question Number 217130 by ArshadS last updated on 02/Mar/25 $$ \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{for}\:\mathrm{every}\:\mathrm{integer}\:\:\mathrm{n}\geqslant\mathrm{2}\:\:\mathrm{the}\:\mathrm{number}\:\:\mathrm{n}^{\mathrm{4}} +\:\mathrm{4}^{{n}} \:\:\mathrm{is} \\ $$$$\mathrm{c}{o}\mathrm{mposite}. \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 217071 by ArshadS last updated on 28/Feb/25 $$\mathrm{Find}\:\mathrm{all}\:\mathrm{two}-\mathrm{digit}\:\mathrm{numbers}\:\mathrm{that}\:\mathrm{are}\:\mathrm{equal}\:\mathrm{to}\:\mathrm{four}\:\mathrm{times}\:\mathrm{the}\:\mathrm{sum}\: \\ $$$$\mathrm{of}\:\mathrm{their}\:\mathrm{digits}.\:\mathrm{Solve}\:\mathrm{this}\:\mathrm{using}\:\mathrm{at}\:\mathrm{least}\:\mathrm{two}\:\mathrm{different}\:\mathrm{methods}\: \\ $$$$\mathrm{and}\:\mathrm{verify}\:\mathrm{your}\:\mathrm{answers}. \\ $$ Answered by som(math1967) last updated on 28/Feb/25 $$\:{x}+\mathrm{10}{y}=\mathrm{4}\left({x}+{y}\right) \\…